1BDJGJD +PVSOBM PG .BUIFNBUJDT · PROPOSITION 1. Suppose that Ù is a simply connected region in C...

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Pacific Journal of Mathematics HYPERBOLIC GEOMETRY IN k-CONVEX REGIONS DIEGO MEJIA AND C. DAVID (CARL)MINDA Vol. 141, No. 2 December 1990

Transcript of 1BDJGJD +PVSOBM PG .BUIFNBUJDT · PROPOSITION 1. Suppose that Ù is a simply connected region in C...

  • Pacific Journal ofMathematics

    HYPERBOLIC GEOMETRY IN k-CONVEX REGIONS

    DIEGO MEJIA AND C. DAVID (CARL) MINDA

    Vol. 141, No. 2 December 1990

  • PACIFIC JOURNAL OF MATHEMATICS

    Vol. 141, No. 2, 1990

    HYPERBOLIC GEOMETRY IN ̂ -CONVEX REGIONS

    DIEGO MEJIA AND DAVID MINDA

    A simply connected region Ω in the complex plane C with smoothboundary convex (k > 0) if k(z9dΩ) > k for allz £ Ω, where k(z,dΩ) denotes the euclidean curvature of dΩ atthe point z. A different definition is used when dΩ is not smooth.We present a study of the hyperbolic geometry of /c-convex regions.In particular, we obtain sharp lower bounds for the density AΩ of thehyperbolic metric and sharp information about the euclidean curvatureand center of curvature for a hyperbolic geodesic in a /c-convex region.We give applications of these geometric results to the family K(k, a)of all conformal mappings / of the unit disk D onto a /c-convex regionand normalized by /(0) = 0 and /'(0) = a > 0. These include precisedistortion and covering theorems (the Bloch-Landau constant and theKoebe set) for the family K(k,a).

    1. Introduction. We study the hyperbolic geometry of certain typesof convex regions called /c-convex regions. It should be emphasizedthat our approach is geometric rather than analytic. Our work is acontinuation of that of Minda ([11], [12], [13], [14]). The paper [11]deals with the hyperbolic geometry of euclidean convex regions, while[12] treats the hyperbolic geometry of spherically convex regions. Areflection principle for the hyperbolic metric was established in [13];this reflection principle leads to a criterion for hyperbolic convexitythat was employed in [14] to give a more penetrating analysis of certainaspects of the hyperbolic geometry of both euclidean and sphericallyconvex regions.

    Roughly speaking, a region Ω in the complex plane C is /c-convex(k > 0), provided k(z9 dΩ) > k for all z e dΩ. Here k(z9 dΩ) denotesthe euclidean curvature of dΩ at the point z. Of course, this onlymakes sense if dΩ is a closed Jordan curve of class C 2 . This conditionis actually sufficient for a region to be /c-convex, but not necessary.Precisely, a region Ω is (euclidean) /c-convex if \a - b\ < 2/k for anypair of distinct points

  • 334 DIEGO MEJIA AND DAVID MINDA

    In his Ph.D. dissertation Mejia [10] investigated the hyperbolic ge-ometry of k-convex regions. Actually, he did even more; he also stud-ied the hyperbolic geometry of /c-convex regions Ω when Ω c P (Pdenotes the Riemann sphere) is k-convex relative to spherical geom-etry or when Ω c D (D is the open unit disk) is /c-convex relative tohyperbolic geometry on D. Independently, Flinn and Osgood [3] in-troduced the notion of hyperbolic /c-convexity for a region Ω c D andstudied some of its properties in the special cases k = 1,2. In subse-quent work we will treat the hyperbolic geometry of both spherically/c-convex and hyperbolically /c-convex regions. Most of the results ofthis paper extend to spherically /c-convex regions by using the samemethods of proof, but for hyperbolically /c-convex regions some newtools are needed, especially in case 0 < k < 2.

    Now, we outline the basic results of the paper. In §2 we present adiscussion of a few elementary euclidean geometrical properties of /c-convex regions that are crucial to our study of the hyperbolic geometryof such regions. In §§3 and 5 we give two different sharp lower boundsfor the density /lΩ(z) of the hyperbolic metric of a /c-convex region Ωin terms of the geometrical quantity SQ(Z), the distance from z to

    l/δςι(z)[2 - kδςι(z)] > 0, actually characterizes /c-convex regions. Asconsequences of these lower bounds, we obtain various results for thefamily K(k, a) of all holomorphic functions / such that / is univalentin D, normalized by /(0) = 0 and /'(0) = a > 0 and /(D) is /c-convex.In particular, we establish a sharp distortion theorem for \f'{z)\ anddetermine the Bloch-Landau constant for the family K(k,a). Thereflection principle for the hyperbolic metric is used to establish acriterion for hyperbolic convexity in /c-convex regions in §6; this resultis best possible for an open disk of radius l//c. Several applicationsof this criterion are given in §§7 and 8. We obtain sharp informationabout the euclidean curvature and center of curvature of a hyperbolicgeodesic in a /c-convex region Ω. The center of curvature must alwayslie in C\Ω, and a sharp lower bound is given for the distance from itto Ω. This leads to both a sharp upper bound on the modulus of thesecond coefficient of a function in K(k, α), together with all extremalfunctions, and an analytic characterization of the family K(k,a). Fork = 0 and a = 1 all of these results for K(k, a) become well-knownfacts for the family K of normalized convex univalent functions.

    The family K(k,a) is a special instance of the family CV{R\,R2),0 < R\ < Z?2> of convex functions of bounded type that was introduced

  • HYPERBOLIC GEOMETRY IN A>CONVEX REGIONS 335

    and studied by Goodman ([4], [5], [6]). Roughly speaking, a normal-ized conformal mapping of D onto a region Ω belongs to CV(R\,R2)if l/R2 < k(z,dΩ) < l/Rι for all z e dΩ. Our family K(k, 1), withk < 1, corresponds to Goodman's class CV(l/k,oo). Some of ourresults for the family K(k, α), such as the determination of the Koebedomain, follow from Goodman's work. On the other hand, we deter-mine the Bloch-Landau constant, a problem not treated by Goodman,and find the sharp upper bound on the second coefficient. Goodmanwas the first to consider the problem of determining bounds on thesecond coefficient and he obtained a first approximation to the sharpbound. In any case, all of our results are obtained as corollaries oftheorems dealing with the hyperbolic geometry of fc-convex regions,an approach quite different from that of Goodman who used onlyelementary methods.

    Finally, we recall a few fundamental facts about the hyperbolic met-ric that we shall use without further comment. For more details thereader should consult ([1], [11], [12], [13], [14]). The density λD ofthe hyperbolic metric on D is λo(z) = 1/(1 - \z\2). If Ω is a regionin C such that C\Ω contains at least two points, then there is a holo-morphic universal covering projection / of D onto Ω. If Ω is simplyconnected, then / is a conformal mapping. The density λςi of the hy-perbolic metric on Ω is determined from λςι(f{z))\f{z)\ = λo(z). It isindependent of the choice of the covering projection. In particular, if/(0) = α, thenλΩ(α) = l/|/'(0)|. For example, if D = {z: \z-a\ < r},then λr>(z) = r/(r2 - \z - a\2). The hyperbolic metric has constantGaussian curvature -4, that is,

    The reader should note that some authors call 2X& the density of thehyperbolic metric; note that 2λςι has Gaussian curvature — 1. We shallrequire two basic properties of the hyperbolic metric. The first is themonotonicity property: If Ω c Δ, then λςι(z) < λΔ(z) for all z e Ωand equality holds at a point if and only if Ω = Δ. The other isthe principle of the hyperbolic metric: If / is holomorphic in D and/(D) c Ω, then λΩ(f(z))\f(z)\ < λD(z) and equality holds at a pointif and only if / is a covering projection onto Ω. When Ω is simplyconnected, equality holds if and only if / is a conformal mappingonto Ω.

    2. Geometric properties of A-convex regions. In this section we intro-duce a measure of the "roundness" of a convex set called fc-convexity.

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    We establish a few basic geometric properties of fc-convex regions thatwill be needed in the remainder of the paper. Most of these resultsare analogs of known facts for convex regions.

    Suppose that k > 0, a, b e C and \a - b\ < 2/k. Then there are twodistinct closed disks D\ and D2 of radius l/k such that α, b e ΘDj (j =1,2). Let Ek[a,b] = D{ ΠD2. Note that the boundary of Ek[a,b]consists of two closed circular arcs Γ\ and Γ2, each of radius l/k andangular length strictly less than π. These arcs have constant euclideancurvature k. We also let E0[a,b] = [a,b], the closed line segmentjoining a and b, and for \a - b\ = 2/k, Ek[a, b] is the closed disk withcenter (a + b)/2 and radius l/k. Then for 0 < k' < k < 2/\a - b\we have Ek,[a,b] c Ek[a,b]. Note that Ek[a,b] is foliated by thecollection of all arcs of constant absolute euclidean curvature k' thatconnect a and b and have angular length less than π, where k1 variesover the interval [0, k].

    DEFINITION. Suppose k e [0,oo). A region Ω c C is called k-convex provided \a - b\ < 2/k for any pair of points a,b e Ω andEk[a,b]cΩ.

    Observe that 0-convex is the same as convex. Henceforth, we shallemploy the term "k-convex" only when k > 0. We will always use"convex" in place of "0-convex". Also, if 0 < k' < k and Ω is k-convex, then Ω is simultaneously Λ:'-convex. In particular, a λ>convexregion is always convex and so simply connected. For k > 0 it iselementary to see that an open disk of radius l/k is Λ -convex, but isnot /c'-convex for any k' > k. Also, if Ωi, . . . , Qn are Λ:-convex, thenP|Ω7 is Λ>convex. Finally, if Ωi c Ω2 C is an increasing sequenceof λ>convex regions, then |J Ω ; is Λ -convex.

    The next result gives an important sufficient condition for a regionto be Λ -convex. Later (Corollary 2 to Theorem 8) we shall see that anyΛ -convex region can be expressed as the increasing union of regionsof the type indicated in the following proposition.

    PROPOSITION 1. Suppose that Ω is a simply connected region in Cbounded by a closed Jordan curve dΩ that is of class C2. Ifk(c, dΩ) >k > 0 for all c e dΩ, then Ω is k-convex.

    Proof. The hypotheses imply that Ω is convex [16, p. 46]. First,we show that if Ί) is any closed disk that is contained in Ω, then theradius of Z) is at most 1/Λ;. Suppose Z) = {z: \z - a\ < r}, wherea e Ω, and δ = δςι(a). Then r < δ so it suffices to show δ < l/k.

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    Suppose that c e then the closed disk with center (a+b)/2 andradius ί/K lies in Ω. From the first part of the proof we obtain XjK <l/k, or k < K. Then Ek[a,b] c Eχ[a,b], so Ek[a,b] is contained inΩ. The other possibility is that K < 2/\a-b\. If Γi and Γ2 are the twocircular arcs of radius \/K that bound Eχ[a,b], then at least one ofthem must meet k for all z € 9Ω, provided dΩ is aC2-closed Jordan curve.

    Recall that if Ω is convex, then for any a e Ω and c e dΩ thehalf-open segment [α, c) c Ω. The next result gives a refinement ofthis fact for /^-convex regions.

    PROPOSITION 2. Suppose Ω is a k-convex region. Then for any a € Ωand c e dΩ, Ek[a, c]\{c} c Ω.

    Proof. Since Ω is convex, the half-open segment [α, c) c Ω. Notethat \a - c\ < 2/k since |α — c| = 2/k would imply that there exista1 eΩ near a and bf eΩ near c with \a! — b'\> 2/k which violates thedefinition of /c-convexity. We first show that intEk[a,c] c Ω. Takecn € [a,c) with cn -* c. Now, cn e Ω, so Ek[a,cn] c Ω for all n.Consequently, intEk[a,c] c U ^ l A ^ ]

    C Ω NOW, we establish thefull result. Select a1 e Ω such that α e (af,c). The first part of theproof gives intEk[a'9c] C Ω. Because Ek[a,c]\{c} c i n t ^ [ α

    ; , c ] , theproof is complete.

  • 338 DIEGO MEJIA AND DAVID MINDA

    COROLLARY. Suppose Ω is a k-convex region. Ifc,d e dΩ, thenintEk[c,d]cΩ.

    Proof. Select dn e Ω_with dn -> d. ΎhenEk[c,dn]\{c} c Ω for all n.This yields Ek[c, d] c Ω, so uΛEk[c9 d] c Ω.

    LEMMA 0. Suppose D is an open disk of radius l/k, B is an opendisk or half plane such that c e dB Γ)dD and B and D are externallytangent at c. If\a -c\< 2/k and aφD, then Ek[a, c]\{c} n B φ 0.

    Proof. Let Γi and Γ2 be the two circular arcs of radius l/k thatbound Ek[a, c], then, except for the point c, one of these two arcs liesentirely outside of Zλ Suppose that Γ\ is this arc. Since Γi cannot betangent to d D at c, it must intersect any disk that is externally tangentto D at c.

    PROPOSITION 3. Suppose Ω is a k-convex region. Assume a € Ω,c e dΩ and \a - c\ = δςι(a). IfD is the open disk of radius l/k thatis tangent to the circle \z - a\ = δςι(a) at c and that contains a in itsinterior, then ΩcD.

    Proof. Let L be the straight line that is tangent to dD at c and letH be the open half-plane determined by L that does not contain α.Then Ω convex implies Ω n H = 0 . If Ω\D Φ 0, then there existsa € Ω\D. Because Ω is open, we may even assume a e Ω\D. Notethat |α — c| < 2/k since Ω is fc-convex. Now, Proposition 2 implies^ [ Λ , C ] \ { C } C Ω, while Lemma 0 gives Ek[a,c]\{c} Π H Φ 0 . Theseresults contradict Ω Π H = 0 . Therefore, Ω c ΰ .

    REMARK. If Ω is convex, α e Ω, c e dΩ, \a-c\ = SQ(S), L is the linetangent to the circle \z-a\ = δςι(a) and H' is the half-plane containingα, then Ω c f f and L is a support line for Ω at c. Proposition 3 givesan analog of this for /:-convex regions: ΩcD and the circle dD ofradius l/k is a support line of constant euclidean curvature k for Ωat c.

    PROPOSITION 4. Suppose Ω is k-convex. Assume a e C\Ω, c e dΩand \a—c\ = δςι{a). IfD is the open disk of radius l/k that is tangent tothe circle \z - a\ = δa(a) at c and that does not meet the open segment(a,c), then ΩcD.

    Proof. Let B = {z: \z - a\ < δΩ(a)} c C\Ω. Then B and D areexternally tangent at c and BnΩ = 0. If Ω\D Φ 0, then there exists

  • HYPERBOLIC GEOMETRY IN fc-CONVEX REGIONS 339

    a E Ω\D and |α — c| < 2/k since Ω is fc-convex. Lemma 0 impliesthat Ek[a,c]\{c}nB φ 0 and Proposition 2 gives Ek[a,c]\{c} c Ω.These facts contradict finΩ = 0.

    3. Lower bound for the density of the hyperbolic metric in a A -convexregion. We obtain a sharp lower bound for the density of the hyper-bolic metric in a ^-convex region. This bound is used to obtain aprecise distortion theorem and a covering theorem for the family ofconformal mappings of the open unit disk D onto Λ -convex regions.These results are generally refinements of known facts for convex re-gions [11].

    THEOREM 1. Suppose Ω is a k-convex region. Then for z e Ω

    A Ω ( Z ) - δΩ(z)[2 - kδa(z)]

    with equality at a point if and only ifΩ is a disk of radius l/k.

    Proof First, assume that D is the open disk with center a and radiusl/k. Then

    ) ~ (l/jfc)2_|z_fl|2

    For z e DyδD{z) = (l/k) -\z-a\ so that

    ; ( λ_ 1 1λD[Z) - δD(z)[l + k\z - a\] δD(z)[2 - kδD(z)]'

    Thus, equality holds for all points in the disk D.Next, consider any fc-convex region Ω. Fix a e Ω. Choose c edΩ

    with \a-c\ = δςι(a). Let D be the disk of radius l/k that is tangent tothe circle \z-a\ = δςι(a) ate and contains α in its interior. Proposition3 gives Ω c ΰ . The monotonicity property of the hyperbolic metricyields λςι(α) > A/>(α) with equality if and only if Ω = D. Since δςι(α) =δj)(α), the preceding inequality together with the work in the aboveparagraph completes the proof.

    COROLLARY 1. Suppose Ω is α k-convex region. Iff is holomorphicin D and /(D) c Ω, then for z e D

    (1 - |z | 2 ) |/ ' (z) | < δΩ(f(z))[2 - kδΩ(f(z))].

    Equality holds at a point if and only ifΩ is a disk of radius l/k and fis a conformal mapping ofΌ onto Ω.

  • 340 DIEGO MEJIA AND DAVID MINDA

    Proof. The principle of the hyperbolic metric gives

    for z G D with equality if and only if / is a conformal mapping of Donto Ω. The theorem yields

    to(/(r))[2-**,(/(*))] " A ί i ( / ( Z ) )

    with equality if and only if Ω is a disk of radius 1 /k. By combining thetwo preceding inequalities and the necessary and sufficient conditionsfor equality, we obtain the desired result.

    DEFINITION. Let K(k, a) denote the family of all holomorphic func-tions / defined on D such that / is univalent, /(0) = 0, /'(0) = a > 0and /(D) is a k-convex region.

    The preceding corollary applied to / e K(k, a) with Ω = /(D) andz = 0 yields a = |/'(0)| < (5Ω(O)[2 - kδQ(O)] = h(δΩ(O)), where h(t) =t(2 - kt). Because h is increasing on [0, l/k] and δΩ(0) < l/k sinceΩ is /c-convex, we obtain a < h(l/k) = l/k whenever / e K{k,a).Also, a = I/A: if and only if /(z) = az.

    EXAMPLE. Set fk(z) = αz/(l - Λ/1 -akz). Then fk e K(k, a) since/(D) is a disk of radius l/k. In fact, fk(-l) = - α / ( l + VI -ak) and/t( l) = α/(l - Λ/1 - α / : ) , so fk{D) is the disk with center y/Y^ak/kand radius l/k. Also, the largest disk contained in fk(D) and centeredat the origin has radius a/(l + y/l — αfe). This example was consideredby Goodman [4].

    As an easy consequence, we obtain the Koebe domain for the familyK(k,a). This result was first obtained by Goodman [4, Theorem 3].

    COROLLARY 2. Suppose f € K(k,a). Then either the closure ofthe disk {w: \w\ < α/(l + Λ/1 -

  • HYPERBOLIC GEOMETRY IN fc-CONVEX REGIONS 341

    4. A characterization of convex and /c-convex regions. We show thatthe inequality for the density of the hyperbolic metric in Theorem1 actually characterizes /c-convex regions. But first we establish theanalogous result for convex regions; we will need this in our proofof the characterization of £>convex regions. The characterization ofconvex regions was demonstrated by Hilditch [8] but not published.Here we present a simple proof that is based on a result of Keogh [9].

    THEOREM 2. Suppose Ω is a hyperbolic region in C. Then Ω isconvex if and only ifλa(z) > l/2δςχ(z) for z G Ω.

    Proof. Minda [11] showed the necessity. Now, we establish thesufficiency. Let f(z) = a§ + a\z + ••• be holomorphic in D with/(D) c Ω. Set σ\{z) = #o + (fli/2)z. The principle of the hyperbolicmetric gives ΛΩ(/(0))|/'(0)| < λD(0) = 1, or \ax\ < l/λQ(a0). Then forz G D we have

    Thus, σ^D) c {w: \w - ao\ < δΩ(ao)} c Ω. It follows [9, Theorem 1]that Ω is convex.

    THEOREM 3. Suppose Ω is a hyperbolic region in C. Then Ω is k-convex if and only ifλςι(z) > l/S&(z)[2 - kδςι(z)] > Ofor z e Ω.

    Proof. We need only establish the sufficiency. The proof is given ina series of steps.

    First, we show that δςι(a) < l/k for a e Ω with equality only if Ωis a disk of radius l/k and center a. Fix a e Ω and set δ = δςι(a),D = {z:\z — a\ < δj. Then D c Ω s o the monotonicity property ofthe hyperbolic metric in conjunction with the hypothesis gives

    Hence, 2 - kδ > 1, or δ < l/k. If equality holds, then λςι(a) = λu{a)which implies Ω = D. The inequality δςι(z) < l/k for z e Ω impliesthat λςι(z) > l/2δςι(z). Therefore, Ω is convex by Theorem 2.

    Second, we prove that if there exist α', έ Έ Ω with \a' - b'\ > 2/k,then there exist a,b eΩ with \a — b\< 2/k and Ek[a,b] (jL Ω. Let mbe the midpoint of the straight line segment [a1, b']; m e Ω because Ωis convex. Since a!,b' G Ω and \a' - bf\ > 2/k, Ω cannot be a disk ofradius l/k. Therefore, the preceding paragraph implies δςι(m) < l/k.

  • 342 DIEGO MEJIA AND DAVID MINDA

    S e l e c t c e < 9 Ω w i t h \ m - c \ = δςι(m). T h e n c e { z : δςι{m) < \ z - m \ <I/A:}. By selecting α, b on [a',bf] Π {z: \z - m\ < I/A:} but close tothe two points in which [a\ b'] meets the circle \z - m\ = I/A:, we caninsure that \a — b\< 2/k and c e Ek[a, b] so that Ek[a, b] (£. Ω.

    Finally, we show that Ω must be /c-convex. Suppose Ω were notA>convex. Then the definition of A>convexity implies that either thereexist points a,b e Ω with \a - b\ > 2/k or points a,b e Ω with\a - b\ < 2/k but Ek[a,b]

  • HYPERBOLIC GEOMETRY IN ^-CONVEX REGIONS 343

    for δ small and positive. But this contradicts our hypothesis thatλa(z) > 11fδςι(z)[2-kδςι(z)] for z e Ω. Consequently, Ω must actuallybe fc-convex.

    5. The Bloch-Landau constant for the family K(k, a). We establish asharp constant lower bound for the density of the hyperbolic metric ofa A:-convex region in terms of the least upper bound for δ&. This lowerbound leads to an easy determination of the Bloch-Landau constantfor the family K(k, a).

    We start by introducing a region that plays the central role in thissection. Fix M e (0, l/k]. Set R = y/M(2 - kM)/k; note thatR= l/k when M = l/k. The number R is determined so that the cir-cle through -RJM and R has radius l/k. Let E = E(M) denotethe λ -convex region inX E^-R^R]. Note that E contains the disk{z: \z\ < M}, but no larger disk, and is contained in the disk D ={z: \z\ < R}. Also, observe that E(l/k) = D. If 2φ is the acute anglethat each of the boundary arcs of E makes with the real axis, thenφ = arctan(M/i?). Essentially this region E was employed in [12] ex-cept that in this reference the region E was normalized by R = 1. Weshall use the results of [12] but for arbitrary R rather than just R = 1;the extension of the results of [12] to arbitrary R is elementary. Theregion E has the feature that circular arcs in E through ±R have con-stant hyperbolic distance from the hyperbolic geodesic (-R,R) andfrom each other. For z e E let y/>(z) denote the hyperbolic distance,relative to Z>, from z to dE. Then the quotient XE/^D can be expressedin terms of y^\ we restate the result here for the general R [12, pp.133-135].

    LEMMA 1. For z eE

    XE{Z) _ πcos[2arctantanh(}>jr>(0) -

    ) 4φ cos [^ arctantanh(yD(0) - γD(z))] ~ 4(P'

    and equality holds if and only ifze (-R9R).

    Next, we extend this inequality from E to certain "triangular" k-convex regions that contain {z: \z\ < M}. Let 2Γ = ^(M) denotethe family of /c-convex regions that contain {z: \z\ < M} and arebounded by three distinct circular arcs each of radius l/k and havingthe property that the full circles are tangent to \z\ = M and contain{z: \z\ < M) in their interior. Observe that if Δ e ZΓy then each ofthe three circular arcs bounding Δ meets the boundary of the disk D

  • 344 DIEGO MEJIA AND DAVID MINDA

    in diametrically opposite points when extended. The following resultis essentially Theorem 2 of [12].

    LEMMA 2. If A e &~, then for z E Δ

    ' 4φ

    Now, we can establish the desired lower bound.

    THEOREM 4. Let Ω be a k-convex region and M = suρ{ 4φR 4 y M(2 - kM) arctan

    Equality holds at a point a e Ω if and only if there is a euclideanmotion T ofC such that Ω = T(E) and a = Γ(0).

    Proof. The proof is an adaptation of that given for the analogousresult for spherically convex regions [12, Theorem 3]. Since Ω isbounded, we actually have M = max{£Ω(z): z e Ω}. Also, M e(0, l/k] since Ω is k-convcx. Take a e Ω with δa(a) = M.

    First, suppose that M = l/k. Then Ω is a disk of radius l/kwith center a (see the proof of Theorem 1), so Ω = T(E), whereT(z) = z + a. Thus,

    with equality if and only if z = a. This establishes the theorem in thespecial case that M = l/k.

    Now, assume that M e (0,1/fc). Because both λ& and

  • HYPERBOLIC GEOMETRY IN fc-CONVEX REGIONS 345

    that Ω c £ . The monotonicity property of the hyperbolic metric inconjunction with Lemma 1 implies that

    λφ) > λE(z) > £- £ π

    4φR

    with strict inequality between the extremes unless z = 0 and Ω = E.The second case occurs when / is not contained in any closed subarc

    of angular length π. Then there exist three points c\9 cι and C3 in /that partition the circle \z\ = M into three subarcs each having angularlength strictly less than π. Let Dj (j = 1,2,3) be the open disk ofradius \/k that is tangent to the circle \z\ = M at Cj and contains 0in its interior. Proposition 3 implies that Ω c Dj, so Ω c f]Dj = Aand Δ € y . Thus, Lemma 2 together with the monotonicity propertyof the hyperbolic metric gives

    This completes the proof.

    The function

    Skit) = Vt(2 - kt) +v ! arctan

    is strictly increasing on [0, l/k] with maximum value g^l/k) = π/4k.Hence, for a e [0, l/k] the equation gk(t) = απ/4 has a unique so-lution M(a) G [0, l/k]. More precisely, we should write Mk(a) inplace of M(a), but we suppress the dependence on k. Note that

    = l/k.

    COROLLARY (Bloch-Landau constant for K(k,a)). Let f e K(k,a).Then either /(D) contains an open disk of radius strictly larger thanM{a), or else f(z) = e~iΘF(eiΘz) for some θ eR, where

    „ , λ /M(a)(2 - kM{a)) 4 uίaF{z) = \ — v Jy ^ M a n hv V j M a n h w Γ Ί Γ 7 Γ T TL O S Ί

    k \2]jM(a)(2-kM(a)) I - z Jbelongs to K(k,a) and maps D conformally onto E(M(a)).

    Proof Set Ω = /(D) and M = max{M(a), then we are done. Suppose that M < M{ά). Then^(M) <gk{M{a)) = απ/4. Now, ΛΩ(0) = 17/(0) = 1/α, so the theorem with

  • 346 DIEGO MEJIA AND DAVID MINDA

    z = /(0) = 0 gives 1/α > π/(4gk{M))y or gk(M) > απ/4. Hence,gk(M) = απ/4, which gives M — M{a). Thus, equality holds in thetheorem at the origin, so Ω is just a rotation of E(M(a)). BecauseF e K(k,a) and maps D conformally onto E(M(a))9 it follows that/(z) = e~iθF(eiΘz) for some θ e R.

    Note that ^ ( ί ) —• t as fc —• 0, so for a fixed value of α, M(α) ->απ/4 as k —> 0. Thus, for α = 1 and k — 0 we obtain Λf(l) = π/4which is the Bloch-Landau constant for the family K of normalizedconvex univalent functions. This result is due to Szegό [17]; also, see[11].

    6. Hyperbolic convexity in k-convex regions. Minda [13] proved thatif Ω φ C is convex and α e Ω, then Ωn{z: \z-a\ < r} is hyperbolicallyconvex as a subset of Ω for any r > 0. Also, if a is in the exterior ofΩ, then this set need not be hyperbolically convex; this is readily seento be true when Ω is a half-plane. Of course, this result also holds for/:-convex regions. In this section we improve this result for /c-convexregions. We show that a can actually lie in the exterior of Ω, providedthere is a restriction on r.

    EXAMPLE. Let D = {z: \z - b\ < I/A:}. Suppose a e C\D, δ =δD(a),r > 0 and δ = kr

    2/(l + y/Y+ΈΦ). Let Γ = dD and Γ ={z: \z - a\ = r}. Then \a - b\2 = (1/fc + δ)2 = (l/k)2 + r2, so thecircles Γ and Γ' are orthogonal. Thus, Γ n D is a hyperbolic geodesicin D and Dn{z: \z — a\ < r] is a hyperbolic half-lane which is triviallyhyperbolically convex as a subset of D. If δ > ks2/(\ + y/l +k2s2),then it is easy to see geometrically that D n {z: \z — a\ < s) is nothyperbolically convex in D.

    THEOREM 5. Suppose Ω w α k-convex region. Let a e C\Ω, J =^Ω(^) ? ^ > 0,i? = {z: |z - a\ < r}, Γ = dR and j denote reflectionin the circle Γ. If δ < kr2/{\ + Vl+k2r2), then j(Ω\R) C Ω. Inparticular, Ω n {z: \z - a\ < r} is hyperbolically convex as a subset ofΩ.

    Proof. Select c e be the open diskof radius l/k that is tangent to the circle \z — a\ = δ at c and doesnot meet the open segment (a,c). Suppose b is the center of D andΓ = dD. Proposition 4 implies that Ω c ΰ .

  • HYPERBOLIC GEOMETRY IN ^-CONVEX REGIONS 347

    First, we consider the case in which δ = kr2/{\ + y/l +k2r2). As inthe preceding example, the circles Γ and P are orthogonal. Considerz e Ω\R and let γ be the circular arc through c, z and j(c). Notethat j(c) e P since P perpendicular to Γ implies that y(P) = P . Infact, c and j{c) are diametrically opposite on the circle P . Let d bethe point in which γ meets Γ. Then it is clear that j(y) = γ since 7fixes d and interchanges the points c and y(c). The point d dividesγ into two subarcs γ\ and ?2, with y\ c i?, 72 C C\i? and 7(72) = JΊIf 5* is the radius of γ, then s > \/k since 7 is inside of Z> and passesthrough diametrically opposite points of P = dD. Let γ' D γ\ be thesubarc of γ from z to c. Proposition 2 implies that γ'\{c} c Ω. Thenj(z) G >(/ n γ2) c yi\{c}, so y(z) e Ω. This proves 7*(Ω\i?) c Ω.This inclusion implies that Ω Π R is hyperbolically convex in Ω ([13,Theorem 6], [14, Proposition 1]).

    The remaining case δ < kr2/(\ + \/l + k2r2) can be reduced to thepreceding case as follows. Select ko > 0 so that

    Then k > ko, so Ω is also ko convex. The first part of the proof appliedwith ko in place of k allows us to conclude that j(Ω\R) c Ω and thatΩ Π R is hyperbolically convex.

    The example prior to the theorem shows that the inequality betweenδ and r in the theorem cannot be improved.

    7. Applications to euclidean curvature. In [14, Theorem 2] Mindaobtained precise information about the location of the center of theeuclidean circle of curvature for a hyperbolic geodesic in a convexregion. In particular, the center of curvature always lies in the com-plement of the region. Now, we determine a sharp relationship be-tween the euclidean curvature and center of curvature for a hyperbolicgeodesic in a fc-convex region.

    THEOREM 6. Suppose that Ω is a k-convex region, γ is a hyperbolicgeodesic in Ω, ZQ G y and the euclidean curvature k(zo,γ) of γ atzo is nonvanishing. Let a denote the euclidean center of curvature andr(zo, y) = l/|fc(zo, y)\ the radius of curvature for y at ZQ. Then a e C\Ωand

    kr(zo,γ)2

    >λ/l+k

    2r(zo,γ)2'

  • 348 DIEGO MEJIA AND DAVID MINDA

    Equality holds if and only ifΩ is an open disk of radius l/k and γ isa circular arc orthogonal to dΩ.

    Proof. We start by showing that equality holds when Ω is a disk ofradius l/k. In this situation the hyperbolic geodesies are the circulararcs and straight line segments that are orthogonal to the boundary.Suppose γ is a circular arc that is perpendicular to r0, center a e C\Ω and the sameunit tangent as γ at z 0 such that δςι(a) < kr

    2/(l + \/l +k2r2). ThenTheorem 5 implies that j(Ω\R) c Ω, where R = {z: \z — a\ < r} and jdenotes reflection in Γ, so [14, Theorem 1] implies k(zo, y) < fc(zo,Γ).Since fc(zo,Γ) < /c(zo,Γo), it follows that ΓQ cannot be the circle ofcurvature for y at ZQ.

    Finally, we determine the form of Ω when equality holds. Sup-pose that Γ is the circle of curvature for γ at z0, has center α, ra-dius r and δςι(a) = kr2/(l + Vl +k2r2). Theorem 5 again impliesthat j(Ω\R) C Ω, so that k(zo,γ) < k(zo,Γ) with equality if andonly if Ω is symmetric about Γ and γ c Γ [14, Theorem 1]. Butk(zQ9γ) = fc(zo,Γ) since Γ is the circle of curvature for γ at ZQ,so Ω must be symmetric about Γ and γ c Γ. Select c e dΩ with\a - c\ = δςι(a). If D is the disk of radius l/k that is tangent to thecircle \z — a\ = δςι{a) at c and does not meet the segment (a,c), thenΩ c f l b y Proposition 4. The fact that equality holds implies that Γ isorthogonal to d D as in the first part of the proof. Because Ω is sym-metric about Γ, j(c) edΩ. Note that j(c) is diametrically opposite con 3D, so \c - j(c)\ = 2/k. But then the corollary to Proposition 2implies that D = intEk[c,j(c)] c Ω. Hence, Ω = Z>, so Ω is a disk ofradius l/k when equality holds.

  • HYPERBOLIC GEOMETRY IN fc-CONVEX REGIONS 349

    The inequality in Theorem 6 is equivalent to each of the followinginequalities:

    \k(z0, γ)\Wk2

    + k(z0,γ)2 + \k(z0,

    COROLLARY 1. Suppose Ω is a k-convex region, y is a hyperbolicgeodesic in Ω and z0 e γ. Then

    Equality implies that Ω is a disk of radius \/k and γ is a circular arcorthogonal to -

    l+Vl+k2r2

    We solve this inequality for 1/r by means of elementary manipulationsand obtain

    " r-δQ(z0)[2-kδa(z0)Y

    COROLLARY 2. Let Ω be a k-convex region. Then for z e Ω

    and

    (ϋ)

    with equality if and only ifΩ, is a disk of radius i/k.

    Proof. First, we show that equality holds for the disk D with center0 and radius l/k. Then λD(z) = k/(l - k

    2\z\2), so that VlogλD(z) =2k2z/(l - k2\z\2). Thus, δD = (l/k) - \z\ gives

    2\z\ 2(1-kδD(z))\V\ogλD(z)\ = [(1 Ik) - \z\W/k) + \z\] δD(z)[2 - kδD(z)\

  • 350 DIEGO MEJIA AND DAVID MINDA

    Similarly, it is elementary to establish equality in (ii) for D.Next, we establish (i). Fix z0 E Ω and set v = VlogAβ(zo). There

    is nothing to prove if v = 0, so we may assume v Φ 0. Set n = vj\v\and let γ be a hyperbolic geodesic through ZQ with normal n at z0.From [15, Formula 19] and Corollary 1 we have

    and equality implies Ω is a disk of radius l/k.Finally, we establish (ii). From Theorem 3 in conjunction with the

    fact that δςι(z) < l/k, we obtain

    a

  • HYPERBOLIC GEOMETRY IN fc-CONVEX REGIONS 351

    The desired result now follows from Corollary 2 to Theorem 6.

    COROLLARY. / / / e K(k,a), then |/"(0)| < 2α^l -

  • 352 DIEGO MEJIA AND DAVID MINDA

    Let φ = -Arg/'(α). Then G = e*g € K(k,(l - |α | 2 ) |/ '(α) |). Thecorollary of Theorem 7 applied to G gives

    \G"(0)/G'(0)\(0).

    The formulas for gf(0) and g"(0) reveal that this inequality is equiv-alent to that stated in the theorem. All that remains is to determinewhen equality holds. If f(z) = e~iθfk{e

    iΘz) for some θ e R, thenequality holds for z = re~iθ, 0 < r < 1. On the other hand, if equalityholds at some point of D, then the proof shows that G, and hence /,maps D onto a disk of radius l/k. Therefore, / must have the forme~wfk{e

    wz) for some θeR.

    COROLLARY 1. Let f be holomorphic and univalent in D and nor-malized by /(0) = 0, /'(0) = a > 0. Then f e K(k,a) if and onlyif

    for zeD.

    Proof. Suppose / e K(k,a). Then the inequality in Theorem 8holds. If we multiply this inequality by \z\, square the resulting in-equality and then simplify, we obtain

    4|z|4 zΓ(z) 2 <- 1 - M2

    This implies the desired result.Conversely, assume that the inequality in the corollary holds. Con-

    sider the path γ: z = z{t) = reιt, t e [0,2π]. The euclidean curvatureof the path / o γ at the point /(z), z e y, is given by

    Consequently, k(f(z),fo γ) > k for all z eγ. Proposition 1 impliesthat /({z: \z\ < r}) is a k-convex region. Because /(D) is an increasingunion of ^-convex regions, it is also /c-convex.

    COROLLARY 2. Suppose f is holomorphic and univalent in D. Then/(D) is k-convex if and only iff maps each subdisk ofD onto a k-convexregion.

    Proof. First, suppose that / maps each subdisk of D onto a /c-convexregion. Then /({z: \z\ < r}) is a /^-convex region for 0 < r < 1, so

  • HYPERBOLIC GEOMETRY IN A>CONVEX REGIONS 353

    /(D) is also /c-convex. Conversely, assume that /(D) is λ -convex.There is no harm in supposing that /(0) = 0 and a = /'(0) > 0,since /c-convexity is invariant under translations and rotations. Webegin by showing that f{{z: \z\ < r}) is a fc-convex region for 0 <r < 1. Corollary 1 implies that if γ: z = z(t) = reu9 t e [0,2π],then k(f(z),fo γ) > k for all z e γ, so Proposition 1 shows thatf({z: \z\ < r}) is a /:-convex region. Now, if Δ is a subdisk of Dwith Δ c D, then there is a conformal automorphism T of D suchthat Γ(Δ) is a disk centered at the origin. Now g = / o T~ι maps Donto a fc-convex region, so the first case shows that /(Δ) = g(T(A))is /c-convex. Finally, if Δ is a subdisk D of that is tangent to the unitcircle, then there is an increasing sequence {An} of subdisks of D withAn c Δ c D for all n and Δ = | JΔ Λ . Since f(An) is fc-convex, it followsthat /(Δ) = \Jf(An) is also λ -convex.

    Added in proof. Professor Wolfram Koepf has pointed out that someof the results of this paper as well as similar results for negtive cur-vature are contained in the paper, E. Peschl, Uber die Krύmmung vonNiveaukurven bei der konformen Abbildung einfachzusammenhάngen-der Gebiete auf das Innere eines Kreises, Math. Ann., 106 (1932),574-594.

    REFERENCES

    [I] L. V. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory,McGraw-Hill, New York, 1973.

    [2] W. Blaschke, Uber den grossten Kreis in einer Konvexen Punktmenge, Jahresber.Deutschen Math. Verein., 23 (1914), 369-374.

    [3] B. Flinn and B. Osgood, Hyperbolic curvature and conformal mapping, Bull.London Math. Soc, 18 (1986), 272-276.

    [4] A. W. Goodman, Convex functions of bounded type, Proc. Amer. Math. Soc,92(1984), 541-546.

    [5] , Convex curves of bounded type, Internat. J. Math. & Math. Sci., 8 (1985),625-633.

    [6] , More on convex functions of bounded type, Proc. Amer. Math. Soc, 97(1986), 303-306.

    [7] H. W. Guggenheimer, Differential Geometry, McGraw-Hill, New York, 1963.[8] J. R. Hilditch, The hyperbolic metric and the distance to the boundary inplane

    domains, unpublished manuscript.[9] F. R. Keogh, A characterisation of convex domains in the plane, Bull. London

    Math. Soc, 8 (1976), 183-185.[10] D. Mejia, The hyperbolic metric in k-convex regions, Ph.D. dissertation, Uni-

    versity of Cincinnati, 1986.[II] D. Minda, Lower bounds for the hyperbolic metric in convex regions, Rocky

    Mountain J. Math., 13 (1983), 61-69.

  • 354 DIEGO MEJIA AND DAVID MINDA

    [12] , The hyperbolic metric and Bloch constants for spherically convex regions,Complex Variables Theory AppL, 5 (1986), 127-140.

    [ 13] , A reflection principle for the hyperbolic metric and applications to geometricfunction theory, Complex Variables Theory AppL, 8 (1987), 129-144.

    [14] , Applications of hyperbolic convexity to euclidean and spherical convexity,J. Analyse Math., 49 (1987), 90-105.

    [15] B. Osgood, Some properties of f"If and the Poincare metric, Indiana Univ.Math. J., 31 (1982), 449-461.

    [16] J. J. Stoker, Differential Geometry, Wiley-Interscience, 1969.[17] G. Szegό, Uber eine Extremalaufgabe aus der Theorie der schlichten Abbildun-

    gen, Sitzungsberichte der Berliner Mathematische Gesellschaft, 22 (1923), 38-47.

    Received February 15, 1988. Research supported in part by NSF Grant No. DMS-8521158.

    UNIVERSIDAD NACIONAL

    APARTADO AέREO 568MEDELLIN, COLOMBIA

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    UNIVERSITY OF CINCINNATI

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  • Pacific Journal of MathematicsVol. 141, No. 2 December, 1990

    Ulrich F. Albrecht, Locally A-projective abelian groups andgeneralizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

    Michel Carpentier, Sommes exponentielles dont la géométrie est très belle:p-adic estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

    G. Deferrari, Angel Rafael Larotonda and Ignacio Zalduendo, Sheavesand functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

    Jane M. Hawkins, Properties of ergodic flows associated to productodometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

    Anthony To-Ming Lau and Viktor Losert, Complementation of certainsubspaces of L∞(G) of a locally compact group . . . . . . . . . . . . . . . . . . . . . . 295

    Shahn Majid, Matched pairs of Lie groups associated to solutions of theYang-Baxter equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

    Diego Mejia and C. David (Carl) Minda, Hyperbolic geometry in k-convexregions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

    Vladimír Müller, Kaplansky’s theorem and Banach PI-algebras . . . . . . . . . . . .355Raimo Näkki, Conformal cluster sets and boundary cluster sets coincide . . . . 363Tomasz Przebinda, The wave front set and the asymptotic support for

    p-adic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383R. F. Thomas, Some fundamental properties of continuous functions and

    topological entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

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    1990Vol.141,N

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